1. Field of the Invention
The present invention relates to two finite difference algorithms and associated models for viscoelastic ink ejection simulation. One algorithm is first-order accurate in time, while the other is second-order accurate in time. The algorithms may be embodied in software, hardware or combination thereof and may be implemented on a computer or other processor-controlled device.
2. Description of the Related Art
Results of ink-jet simulation have been very useful in the design of piezoelectric ink-jet print heads. A practical ink-jet simulation may be carried out using an analytical tool such as an equivalent circuit that receives as an input the dynamic voltage to be applied to a piezoelectric actuator and simulates the ink behavior under the influence of the ink cartridge, supply channel, vibration plate, and actuator. That is, from the input voltage and an ink flow rate, the equivalent circuit calculates an inflow pressure that drives computational fluid dynamics (CFD) code. The CFD code then solves the governing partial differential equations, i.e., the incompressible Navier-Stokes equations for two-phase flows, for fluid velocity, pressure and interface position, and feeds back the ink flow rate to the equivalent circuit. The sequence is repeated as long as needed.
The dynamics of fluid flow through the ink nozzle, coupled to surface tension effects along the ink-air interface and boundary conditions along the wall, act to determine the shape of the interface as it moves. Designing the CFD code largely involves taking into account the dynamically changing ink-air interface, which is quite challenging.
One method that has been used to model the ink-air interface is the volume of fluid method (VOF). VOF performs fairly well with regard to mass conversation but is not so accurate on surface tension aspects of fluid flow, especially when the ink droplet is smaller than 5 pico liters. Considering that the capability of ejecting ultra small ink droplets is essential for any photo quality ink-jet printer today, VOF gave way to an improved modeling method which included the level set method to more accurately capture the ink-air interface in CFD simulations. Since there is a mathematical relation between the level set and the interface curvature, and hence the surface tension, the level set method excels whenever surface tension is important.
One way to solve the level set equations is by finite element analysis. However, because this solving method usually results in a serious mass conservation problem, finite difference analysis is usually the best choice among numerical schemes to be used with the level set method.
The present invention continues along this technological path of using finite difference analysis in ink ejection simulation. In so doing, the present invention provides substantial improvements in finite difference algorithms performed on rectangular grids for viscoelastic ink ejection simulations.